#88062

order	416
isoclinism	isoclinic class of #6104
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₁₃ ⋊ #108

Subgroups

	subgroup	G/H
center	C₄	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₃
Fitting	C₁₃×M₁₆	C₂
socle	C₂×C₁₃	C₄⋊C₄
prime radical	C₂²×C₁₃	Q₈
cosocle	C₂×C₄×C₁₃	C₂²
hypercenter	M₁₆	D₁₃
hypocenter	C₁₃	#108
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 208

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2,4,6,8,10,12,13,11,9,7,5,3)(14,16,17,25,19,26,18,27)(15,20,21,24,23,28,22,29);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15,18,22,19,23,17,21)(16,24,25,28,26,29,27,20)

Polycyclic presentation:

2^5,13|a^2=d,b^2=c,c^2=e,d^2=e,b^a=bd,f^a=f^12,d^b=de

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718409, #11718410, #11718415, …
maximal normal subgroup of	#11718409, #11718410, #11718415, …
maximal quotient of	#11718071, #11718072, #11718150, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 29);